A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains

TL;DR

This work introduces a thermodynamically consistent dual-potential framework to model anisotropic inelasticity at finite strains by augmenting neural networks with invariants and physics-based constraints. It combines Input Convex Neural Networks and Input Monotonic Neural Networks to represent the Helmholtz free energy and the dual potential, respectively, and employs Liquid Neural Networks to update internal variables, enabling stable training without purely convex assumptions. Demonstrations at material-point and structural scales show accurate, robust predictions for isotropic and anisotropic responses, outperforming purely recurrent baselines and maintaining stability beyond training data. The approach is provided as open-source, highlighting its potential for engineering applications and further extensions to other inelastic phenomena.

Abstract

We propose a complement to constitutive modeling that augments neural networks with material principles to capture anisotropy and inelasticity at finite strains. The key element is a dual potential that governs dissipation, consistently incorporates anisotropy, and-unlike conventional convex formulations-satisfies the dissipation inequality without requiring convexity. Our neural network architecture employs invariant-based input representations in terms of mixed elastic, inelastic and structural tensors. It adapts Input Convex Neural Networks, and introduces Input Monotonic Neural Networks to broaden the admissible potential class. To bypass exponential-map time integration in the finite strain regime and stabilize the training of inelastic materials, we employ recurrent Liquid Neural Networks. The approach is evaluated at both material point and structural scales. We benchmark against recurrent models without physical constraints and validate predictions of deformation and reaction forces for unseen boundary value problems. In all cases, the method delivers accurate and stable performance beyond the training regime. The neural network and finite element implementations are available as open-source and are accessible to the public via https://doi.org/10.5281/zenodo.17199965.

Figures (19)

• Figure 1: Schematic illustration of possible constructions for a function $\varphi(\Sigma)$ satisfying Inequality \ref{['eq:dissipation_1D']}. The function $\omega$ is convex, non-negative, and zero-valued at the origin, whereas $\zeta$ is monotonically increasing and zero-valued with respect to $\omega$. Their composition $(\zeta \circ \omega)$ ensures that the sign of the subgradient $\partial_\Sigma \varphi$ coincides with the sign of $\Sigma$ in the negative and positive regimes, respectively.
• Figure 2: Schematic illustration of the composition network $\mathcal{N}_\circ$. The blue part represents the Input Convex Neural Network $\mathcal{N}_c$, which is convex in $\mathbf{y}$ but not in $\mathbf{x}$. Each layer takes as input the output of the previous layer and of the parallel network in gray carrying $\mathbf{x}$ as well as $\mathbf{y}$ itself. Its final output $\mathbf{z}_C$ highlighted in red serves as the input $\mathbf{s}$ to the Input Monotonic Neural Network $\mathcal{N}_m$ shown in orange, whose architecture mirrors that of $\mathcal{N}_c$. To guarantee a non-negative $\mathbf{z}_C$, the activation between $\mathbf{z}_{C-1}$ and $\mathbf{z}_C$ is chosen as ReLU.
• Figure 3: Motion from the reference configuration $\mathscr{B}_0$ to the current configuration $\mathscr{B}$. An infinitesimal material element is described by the deformation gradient $\bm{F}$, which admits a multiplicative decomposition into elastic $\bm{F}_e$ and inelastic $\bm{F}_i$ parts. Due to the non-uniqueness, one may equivalently write $\bm{F}=\bm{F}_{e*}\,\bm{F}_{i*}$ with $\bm{F}_{e*}=\bm{F}_e\,\bm{Q}^T$ and $\bm{F}_{i*}=\bm{Q}\,\bm{F}_i$ with $\bm{Q}\in\mathrm{SO}(3)$. From the polar decompositions of $\bm{F}_i$ and $\bm{F}_{i*}$, it follows that $\bm{R}_i^*=\bm{Q}\bm{R}_i$. Noteworthy, the shown quantities referred to one intermediate configuration share their eigenvalues with the corresponding quantities in any other intermediate configuration.
• Figure 4: Illustration of the normalization functions for the isotropic $\mathcal{S}_\psi$ and anisotropic $\mathcal{A}_\psi$ sets for the Helmholtz free energy $\psi$.
• Figure 5: Schematic of time discretizations of the ordinary differential equation $\dot{y} = A(y)\,y$ with $A(y) = r(1 - y/k)$, where $r = 0.7$ and $k = 10$. The exponential integrator with time step $\Delta t = 1.5$ shows the explicit scheme $y_{n+1} = \exp(\Delta t\,A(y_n))\,y_n$ and the implicit scheme $y_{n+1} = \exp(\Delta t\,A(y_{n+1}))\,y_n$, the latter must be solved numerically. The Liquid Neural Network (LiNN) approximates $y_{n+1}$ by minimizing the loss $\mathcal{L}=(y_{n+1} - \exp(\Delta t\,A(y_{n+1}))\,y_n)^2$. Within the training domain, the LiNN attains accuracy comparable to the implicit scheme while remaining explicit; outside this domain, predictive reliability is not ensured.